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• 2010 USAMO Problems/Problem 4
  We know that angle <math>BIC = 135^{\circ}</math>, as the other two angles in triangle <math>BIC</math> add to <math>45^{\circ}</math>. Assume markangle(Label("$\scriptstyle{\frac{\pi}{4} - \frac{\theta}{2}}$"), radius=40, I, C, B);
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2007 AMC 8 Problems/Problem 25
  ...rt will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regio ...> or <math>9\pi</math>. Note that the area of the whole circle is <math>36\pi</math>.
  3 KB (510 words) - 10:08, 18 June 2024
• 2010 AMC 12B Problems/Problem 18
  ...and either circle's center (symmetry, you chose!). The intersection of the two circles should form a geometrical lens shape. By sectors, <cmath>f(\theta)=\frac{\theta}{2\pi}\cdot2\pi=\theta</cmath>
  6 KB (1,105 words) - 13:39, 9 January 2024
• 2004 AMC 8 Problems
  Find the number of two-digit positive integers whose digits total <math>7</math>. ...of the five numbers in a list is <math>54</math>. The average of the first two
  13 KB (1,860 words) - 18:51, 13 June 2024
• 2006 AMC 8 Problems
  ...{4}</math> revolutions. In what direction does the spinner point after the two moves? The letter T is formed by placing two <math> 2 \times 4 </math> inch rectangles next to each other, as shown. Wha
  16 KB (2,215 words) - 19:18, 10 April 2024
• 2010 AMC 10B Problems/Problem 20
  Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tange ...area of the first circle is <math>(\frac{\sqrt{3}}{6})^2 \cdot \pi =\frac{\pi}{12}</math>.
  3 KB (483 words) - 18:41, 4 May 2024
• 2004 JBMO Problems/Problem 4
  ...have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the tr Prove that there are two more black triangles that white ones.
  2 KB (451 words) - 01:03, 23 December 2018
• 2010 AMC 10B Problems/Problem 19
  A circle with center <math>O</math> has area <math>156\pi</math>. Triangle <math>ABC</math> is equilateral, <math>\overline{BC}</math The formula for the area of a circle is <math>\pi r^2</math> so the radius of this circle is <math>\sqrt{156}.</math>
  3 KB (443 words) - 18:34, 4 May 2024
• 2010 AMC 10B Problems/Problem 16
  ...c{\pi}{18} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{2\pi}{9}</math> ...f by the two radii - area of the triangle with sides on the square and the two radii).
  3 KB (544 words) - 20:54, 24 March 2024
• 2011 AMC 12A Problems
  ...nts per text message sent, plus <math>10</math> cents for each minute used over <math>30</math> hours. In January Michelle sent <math>100</math> text messa ...ber of successful free throws was one more than their number of successful two-point shots. The team's total score was <math>61</math> points. How many fr
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 10
  ...mes \frac{1}{6} \times \frac{1}{6} = \frac{2}{36}</math>. The sum of these two probabilities now gives the final answer: <math>\frac{1}{36} + \frac{2}{36}
  2 KB (248 words) - 14:51, 5 May 2021
• 1958 AHSME Problems
  Which of these four numbers <math> \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{-1}\cdot \sqrt{(.09)^{-1}}< The sum of two numbers is <math> 10</math>; their product is <math> 20</math>. The sum of
  25 KB (3,872 words) - 14:21, 20 February 2020
• 2011 AMC 10A Problems
  ...math>¢ per text message sent, plus <math>10</math>¢ for each minute used over <math>30</math> hours. In January Juan sent <math>100</math> text messages ...ber of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws d
  13 KB (1,903 words) - 18:09, 19 April 2021
• 2011 AMC 10B Problems/Problem 18
  ...cmath>\tan (\theta) = \frac{3}{x}</cmath> YAY!!! We have two equations for two variables... that are relatively difficult to deal with. Well, we'll try to ...{6 - 3 \sqrt{3}} = 2 + \sqrt{3}</math>, <math>\theta = \frac{5 + 12n}{12} \pi</math>, where <math>n</math> is any integer. Converting to degrees, we have
  5 KB (782 words) - 14:29, 1 April 2024
• 2011 AMC 12B Problems
  ...went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and ...integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the c
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2002 AMC 10B Problems/Problem 5
  ...d \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi </math> A line going through the centers of the two smaller circles also goes through the diameter. The length of this line wit
  2 KB (247 words) - 17:30, 5 January 2021
• 2011 AIME I Problems
  ...um of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can b Suppose <math>x</math> is in the interval <math>[0,\pi/2]</math> and <math>\log_{24 \sin x} (24 \cos x) = \frac{3}{2}</math>. Fin
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2011 AMC 12B Problems/Problem 8
  ...)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math> The curves of the track are semicircles, but since there are two of them, we can consider both of the at the same time by treating them as a
  2 KB (389 words) - 19:35, 7 August 2023
• 2011 AIME I Problems/Problem 9
  Suppose <math>x</math> is in the interval <math>[0, \pi/2]</math> and <math>\log_{24\sin x} (24\cos x)=\frac{3}{2}</math>. Find <ma There are now two ways to finish this problem.
  2 KB (330 words) - 20:47, 10 December 2023
• 2011 AIME I Problems/Problem 14
  ...7}}</math> are parallel, and its side length is the distance between these two lines. However, this is given to be the side length of the octagon, or <mat ...nt-and-a-line/ point to line distance formula], the distance between these two lines is <cmath>\frac{|c_{2}-c_{1}|}{\sqrt{a^{2}+b^{2}}}=\frac{2m\left(1+\s
  8 KB (1,344 words) - 18:39, 9 February 2023

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